{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0001","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4204c53a9b2d54b703d4fdd308339cf200b847fefbb82d8f31da9beba9d8b29","family_id":"qf-mod32-asconv-qualification","seed":202611014001,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[1,2,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[1, 2, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 1, Yes, Yes], [1, 1/2, 2, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 1, Yes, Yes], [1, 1/2, 2, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0002","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5264e521177925e3946b7ce5d77dd5f20d90500f217c7727f9870c086642034e","family_id":"qf-mod32-asconv-qualification","seed":202611014002,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[2,3,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[2, 3, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 2, Yes, Yes], [1, 1/2, 3, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 2, Yes, Yes], [1, 1/2, 3, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0003","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2e431e20863b1a27c35954b17a58ac514345099aa664a3ffc1a23fcac11eca6","family_id":"qf-mod32-asconv-qualification","seed":202611014003,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[3,4,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[3, 4, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 3, Yes, Yes], [1, 1/2, 4, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 3, Yes, Yes], [1, 1/2, 4, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0004","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5fa2b059d91d750ec13f4721499e2b4d37509fc80fc58f31a2e9fcb3e06d630","family_id":"qf-mod32-asconv-qualification","seed":202611014004,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[4,5,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[4, 5, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 4, Yes, Yes], [1, 1/2, 5, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 4, Yes, Yes], [1, 1/2, 5, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0005","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84438f65ceccd69d5b29d3aa77ac08f6b20264c228ed47de689a8d82ae0a78ce","family_id":"qf-mod32-asconv-qualification","seed":202611014005,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[5,6,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[5, 6, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 5, Yes, Yes], [1, 1/2, 6, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 5, Yes, Yes], [1, 1/2, 6, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0006","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a5c43f35b7ff9109bc1fde3b6a590dbf436fc2b1c1cb1306c28d112b892b45e","family_id":"qf-mod32-asconv-qualification","seed":202611014006,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[6,7,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[6, 7, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 6, Yes, Yes], [1, 1/2, 7, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 6, Yes, Yes], [1, 1/2, 7, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0007","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c504b6707e1bdbdc00ace2ab72695dfaca71bed41622eb88e30bd1abe1259683","family_id":"qf-mod32-asconv-qualification","seed":202611014007,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[7,8,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[7, 8, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 7, Yes, Yes], [1, 1/2, 8, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 7, Yes, Yes], [1, 1/2, 8, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0008","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a81c3d975bc95df8dedee7da7fb4564139ae3621c1d0bb8bb547a08e2b964f3d","family_id":"qf-mod32-asconv-qualification","seed":202611014008,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[8,9,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[8, 9, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 8, Yes, Yes], [1, 1/2, 9, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 8, Yes, Yes], [1, 1/2, 9, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0009","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2de8a7f0d46320f8635456ad2b1f81386769c92b65bdf4f201a1eae433415f0","family_id":"qf-mod32-asconv-qualification","seed":202611014009,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[9,10,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[9, 10, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 9, Yes, Yes], [1, 1/2, 10, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 9, Yes, Yes], [1, 1/2, 10, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0010","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4b05851f9083843d104699f04f7ad080e7ae44942c6942f94e8112e51604727","family_id":"qf-mod32-asconv-qualification","seed":202611014010,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[10,11,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[10, 11, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 10, Yes, Yes], [1, 1/2, 11, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 10, Yes, Yes], [1, 1/2, 11, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0011","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d85e7f008ea1d81296b7236ecd46f8b5293097c5d2562c11cadd3e7dd86738ab","family_id":"qf-mod32-asconv-qualification","seed":202611014011,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[11,12,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[11, 12, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 11, Yes, Yes], [1, 1/2, 12, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 11, Yes, Yes], [1, 1/2, 12, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0012","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b3fd14af4458c2f497297ee8f9bae9af761378ef404588a5e428f080e4d554f","family_id":"qf-mod32-asconv-qualification","seed":202611014012,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[12,13,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[12, 13, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 12, Yes, Yes], [1, 1/2, 13, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 12, Yes, Yes], [1, 1/2, 13, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0013","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6f0b20398f7f777aa6a14efff3a9837760970751c47823bbb43448f9003d409","family_id":"qf-mod32-asconv-qualification","seed":202611014013,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[13,14,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[13, 14, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 13, Yes, Yes], [1, 1/2, 14, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 13, Yes, Yes], [1, 1/2, 14, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0014","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9049a0d129f15182c5ae0c0ea94db397901a2542c85f168d1148a232fd256275","family_id":"qf-mod32-asconv-qualification","seed":202611014014,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[14,15,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[14, 15, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 14, Yes, Yes], [1, 1/2, 15, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 14, Yes, Yes], [1, 1/2, 15, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0015","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85602bde04b439b52b55de3abeeacd40f09486ec104e3d4bcc4b93c9169a79ad","family_id":"qf-mod32-asconv-qualification","seed":202611014015,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[15,16,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[15, 16, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 15, Yes, Yes], [1, 1/2, 16, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 15, Yes, Yes], [1, 1/2, 16, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0016","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9a43bd0862c576df59a19e64bc5ef57b0e3ad99798e1b3246b783c52badc8b3","family_id":"qf-mod32-asconv-qualification","seed":202611014016,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[16,17,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[16, 17, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 16, Yes, Yes], [1, 1/2, 17, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 16, Yes, Yes], [1, 1/2, 17, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0017","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68b1d055cf149cf0805af72d76f1873098d7396ccc5dd22a512d0f1bdd2ca036","family_id":"qf-mod32-asconv-qualification","seed":202611014017,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[17,18,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[17, 18, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 17, Yes, Yes], [1, 1/2, 18, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 17, Yes, Yes], [1, 1/2, 18, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0018","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4dc1672203f9955af764f6dce9a35def08a1a623f84b4a6eaaff7c69a861210","family_id":"qf-mod32-asconv-qualification","seed":202611014018,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[18,19,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[18, 19, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 18, Yes, Yes], [1, 1/2, 19, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 18, Yes, Yes], [1, 1/2, 19, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0019","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"790dcf3e7e0682d472f74c52ed8b79e882be271fd06f942062822f6666eb00e5","family_id":"qf-mod32-asconv-qualification","seed":202611014019,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[19,20,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[19, 20, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 19, Yes, Yes], [1, 1/2, 20, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 19, Yes, Yes], [1, 1/2, 20, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0020","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c832b6ac07237a289dd7c1b81fa9f22751bf64c807170d9226a3cbc06662612d","family_id":"qf-mod32-asconv-qualification","seed":202611014020,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[20,21,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[20, 21, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 20, Yes, Yes], [1, 1/2, 21, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 20, Yes, Yes], [1, 1/2, 21, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0021","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7f672e861b6dd3115fc3efaa49a39dead3d0634b127689c1a9614c50a03087a","family_id":"qf-mod32-asconv-qualification","seed":202611014021,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[21,22,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[21, 22, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 21, Yes, Yes], [1, 1/2, 22, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 21, Yes, Yes], [1, 1/2, 22, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0022","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79617205a1ae1c2180c9de78e9077ddbd58fc4005d98fe7fa28cdefc2bc82d58","family_id":"qf-mod32-asconv-qualification","seed":202611014022,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[22,23,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[22, 23, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 22, Yes, Yes], [1, 1/2, 23, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 22, Yes, Yes], [1, 1/2, 23, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0023","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe4efaa397c79b1dbfd5ef53486bff6d6a65a04373a3ead17c3362ff08ecb1fa","family_id":"qf-mod32-asconv-qualification","seed":202611014023,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[23,24,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[23, 24, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 23, Yes, Yes], [1, 1/2, 24, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 23, Yes, Yes], [1, 1/2, 24, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0024","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2ea652b2ebd6af6acc43f01e2e8dc821880cc7081253f294f91f9b7587ebbec","family_id":"qf-mod32-asconv-qualification","seed":202611014024,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[24,25,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[24, 25, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 24, Yes, Yes], [1, 1/2, 25, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 24, Yes, Yes], [1, 1/2, 25, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0025","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36b968044af2dd07572b7ce4e2b640cb75c5c26927eef1d3ec1c253bac196e36","family_id":"qf-mod32-asconv-qualification","seed":202611014025,"semantic_template":"qf-mod32-asconv-qualification/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[25,26,null]}"},"fields":{"title":"Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[25, 26, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (atom_rows,exceptional_mass,positive_mass_failures,almost_sure?). Each row is [atom,mass,settling_index,positive_mass?,zero_mass_or_convergent?] in outcome order.","answer":{"kind":"expression","value":"([[0, 1/2, 25, Yes, Yes], [1, 1/2, 26, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)","explanation":"Sum masses of all null-settling atoms for the exceptional mass and list the positive-mass failures separately. A null declaration on a zero-mass atom does not obstruct the almost-sure conclusion. This is a full declared tail model, not an inference from finite observations. The exact result, retaining the requested order, is ([[0, 1/2, 25, Yes, Yes], [1, 1/2, 26, Yes, Yes], [2, 0, null, No, Yes]], 0, [], Yes)."},"skills":["Only Positive-Mass Nonconvergent Atoms Obstruct This Almost-Sure Model"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0026","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e218486dfe9e007e99620742f802285fdcf85c989d571bb2b6bfbae0f433a0ec","family_id":"qf-mod32-asconv-settling-index","seed":202611014026,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[1,2,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[1, 2, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 1, Yes, Yes], [1, 1/2, 2, Yes, Yes]], 2, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 1, Yes, Yes], [1, 1/2, 2, Yes, Yes]], 2, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0027","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d08ccadb82e43bca2c1a11f94247767a7e506166cb979dc6acaa5a257f15bb2a","family_id":"qf-mod32-asconv-settling-index","seed":202611014027,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[2,3,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[2, 3, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 2, Yes, Yes], [1, 1/2, 3, Yes, Yes]], 3, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 2, Yes, Yes], [1, 1/2, 3, Yes, Yes]], 3, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0028","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"975c69c1d184750f5e261582f6edf84f628fbaf2ef90148c7bd2e973f8a1f34b","family_id":"qf-mod32-asconv-settling-index","seed":202611014028,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[3,4,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[3, 4, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 3, Yes, Yes], [1, 1/2, 4, Yes, Yes]], 4, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 3, Yes, Yes], [1, 1/2, 4, Yes, Yes]], 4, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0029","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9fb18fea3a409f7ad69632e8e68ad77b19e21260b4b23cdb35953c3722ee0f5d","family_id":"qf-mod32-asconv-settling-index","seed":202611014029,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[4,5,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[4, 5, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 4, Yes, Yes], [1, 1/2, 5, Yes, Yes]], 5, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 4, Yes, Yes], [1, 1/2, 5, Yes, Yes]], 5, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0030","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3339d1d7814d68f708da10de94af01820538d2be46931b2c3266ca0c502d6899","family_id":"qf-mod32-asconv-settling-index","seed":202611014030,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[5,6,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[5, 6, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 5, Yes, Yes], [1, 1/2, 6, Yes, Yes]], 6, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 5, Yes, Yes], [1, 1/2, 6, Yes, Yes]], 6, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0031","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"526d377f42b5203c4fabbae0dd04c9599cb373d846356cd82ff4ee2149b9fa1f","family_id":"qf-mod32-asconv-settling-index","seed":202611014031,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[6,7,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[6, 7, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 6, Yes, Yes], [1, 1/2, 7, Yes, Yes]], 7, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 6, Yes, Yes], [1, 1/2, 7, Yes, Yes]], 7, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0032","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5947e6502e82220e56d45c7bdd1e05cd0386115beb5f1b7fccbc6c97e6d1971","family_id":"qf-mod32-asconv-settling-index","seed":202611014032,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[7,8,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[7, 8, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 7, Yes, Yes], [1, 1/2, 8, Yes, Yes]], 8, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 7, Yes, Yes], [1, 1/2, 8, Yes, Yes]], 8, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0033","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8fa8b417999390c930e3a75a1e8f6ba617316362bd34a73b95f5e0c5a426f95","family_id":"qf-mod32-asconv-settling-index","seed":202611014033,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[8,9,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[8, 9, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 8, Yes, Yes], [1, 1/2, 9, Yes, Yes]], 9, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 8, Yes, Yes], [1, 1/2, 9, Yes, Yes]], 9, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0034","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b9606280d484b1513ca93deecb61559b26885b406ca2c7855c973dc5c2374b0","family_id":"qf-mod32-asconv-settling-index","seed":202611014034,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[9,10,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[9, 10, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 9, Yes, Yes], [1, 1/2, 10, Yes, Yes]], 10, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 9, Yes, Yes], [1, 1/2, 10, Yes, Yes]], 10, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0035","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d835b9cce78c0e3966bc9f1bdb560ef8e1a0bcf7f5b4e0518c70c322091e66a2","family_id":"qf-mod32-asconv-settling-index","seed":202611014035,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[10,11,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[10, 11, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 10, Yes, Yes], [1, 1/2, 11, Yes, Yes]], 11, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 10, Yes, Yes], [1, 1/2, 11, Yes, Yes]], 11, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0036","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"054b646a9f9112cf4da2aa750704a63427614a44da4b29b2a8d95d868e0cde9b","family_id":"qf-mod32-asconv-settling-index","seed":202611014036,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[11,12,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[11, 12, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 11, Yes, Yes], [1, 1/2, 12, Yes, Yes]], 12, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 11, Yes, Yes], [1, 1/2, 12, Yes, Yes]], 12, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0037","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1722ff35c8c3136df303c241bdee50608257a4cf9498c1ad88316b57e3b24f1","family_id":"qf-mod32-asconv-settling-index","seed":202611014037,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[12,13,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[12, 13, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 12, Yes, Yes], [1, 1/2, 13, Yes, Yes]], 13, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 12, Yes, Yes], [1, 1/2, 13, Yes, Yes]], 13, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0038","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0724d82b2fe5f11d6e6e3eb482b704bb4eec86170ae6023c766f3f7ddcda34c7","family_id":"qf-mod32-asconv-settling-index","seed":202611014038,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[13,14,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[13, 14, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 13, Yes, Yes], [1, 1/2, 14, Yes, Yes]], 14, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 13, Yes, Yes], [1, 1/2, 14, Yes, Yes]], 14, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0039","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39d7ad90868b334a6b09a973448360c03dcae2f53f429247ffe3421ef5a2492d","family_id":"qf-mod32-asconv-settling-index","seed":202611014039,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[14,15,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[14, 15, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 14, Yes, Yes], [1, 1/2, 15, Yes, Yes]], 15, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 14, Yes, Yes], [1, 1/2, 15, Yes, Yes]], 15, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0040","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03f49bc8329cb46b2681c1dddec6604ab64cb7b67ea7e3abd1f79badf7187dae","family_id":"qf-mod32-asconv-settling-index","seed":202611014040,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[15,16,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[15, 16, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 15, Yes, Yes], [1, 1/2, 16, Yes, Yes]], 16, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 15, Yes, Yes], [1, 1/2, 16, Yes, Yes]], 16, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0041","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2872f5042c15d5459d4c2e98de17a889cbd7e33f86d0bdbc00d1515877fd207","family_id":"qf-mod32-asconv-settling-index","seed":202611014041,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[16,17,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[16, 17, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 16, Yes, Yes], [1, 1/2, 17, Yes, Yes]], 17, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 16, Yes, Yes], [1, 1/2, 17, Yes, Yes]], 17, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0042","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50738ac3ece900b6c0ef75ef4d30650a29489b4fe3643dbbbbbad2db0b794d51","family_id":"qf-mod32-asconv-settling-index","seed":202611014042,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[17,18,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[17, 18, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 17, Yes, Yes], [1, 1/2, 18, Yes, Yes]], 18, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 17, Yes, Yes], [1, 1/2, 18, Yes, Yes]], 18, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0043","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93d441163cdea65558242789c33f24b740f0391d6b23029282103a98471f4f61","family_id":"qf-mod32-asconv-settling-index","seed":202611014043,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[18,19,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[18, 19, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 18, Yes, Yes], [1, 1/2, 19, Yes, Yes]], 19, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 18, Yes, Yes], [1, 1/2, 19, Yes, Yes]], 19, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0044","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbb11ec424489604d16dc782294cddfa12459f2480874642ec13835e5c868d18","family_id":"qf-mod32-asconv-settling-index","seed":202611014044,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[19,20,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[19, 20, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 19, Yes, Yes], [1, 1/2, 20, Yes, Yes]], 20, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 19, Yes, Yes], [1, 1/2, 20, Yes, Yes]], 20, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0045","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72cea4a4014cd8ad3c3cbe95a88bc1d49f71402207572a323de37bb36269238b","family_id":"qf-mod32-asconv-settling-index","seed":202611014045,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[20,21,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[20, 21, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 20, Yes, Yes], [1, 1/2, 21, Yes, Yes]], 21, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 20, Yes, Yes], [1, 1/2, 21, Yes, Yes]], 21, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0046","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85f09944cc3dc74eabfbacd9680f0f981515b12c35a6c4b2314b413beda13e07","family_id":"qf-mod32-asconv-settling-index","seed":202611014046,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[21,22,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[21, 22, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 21, Yes, Yes], [1, 1/2, 22, Yes, Yes]], 22, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 21, Yes, Yes], [1, 1/2, 22, Yes, Yes]], 22, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0047","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74ce21843ed51082522a884a37e840107a9896eec8151f28b2d0252f744f73e5","family_id":"qf-mod32-asconv-settling-index","seed":202611014047,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[22,23,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[22, 23, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 22, Yes, Yes], [1, 1/2, 23, Yes, Yes]], 23, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 22, Yes, Yes], [1, 1/2, 23, Yes, Yes]], 23, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0048","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60247379b9ae9a7263a6f35fce96e5630ef5b3b857fae85c93429708c2d7907b","family_id":"qf-mod32-asconv-settling-index","seed":202611014048,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[23,24,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[23, 24, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 23, Yes, Yes], [1, 1/2, 24, Yes, Yes]], 24, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 23, Yes, Yes], [1, 1/2, 24, Yes, Yes]], 24, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0049","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"739c818d2918bd02dd7dd27ae35ab2f041dffa4e8b7a966a4f0692b01395b2e3","family_id":"qf-mod32-asconv-settling-index","seed":202611014049,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[24,25,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[24, 25, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 24, Yes, Yes], [1, 1/2, 25, Yes, Yes]], 25, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 24, Yes, Yes], [1, 1/2, 25, Yes, Yes]], 25, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0050","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc038d857e7da8eedc473a3b9c81119aa4fa5de04488c41c59a240cb1a497318","family_id":"qf-mod32-asconv-settling-index","seed":202611014050,"semantic_template":"qf-mod32-asconv-settling-index/{\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[25,26,null]}"},"fields":{"title":"A Finite Number of Positive Atoms Gives a Maximum Settling Index","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[25, 26, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Give (positive_mass_rows,maximum_settling_index,almost_sure?,certificate). The certificate is uniform_finite_atom_tail_after_maximum or positive_mass_nonconvergent_atom.","answer":{"kind":"expression","value":"([[0, 1/2, 25, Yes, Yes], [1, 1/2, 26, Yes, Yes]], 26, Yes, uniform_finite_atom_tail_after_maximum)","explanation":"Retain the rows of positive-mass atoms only. If all settle, take the maximum of their declared indices for a common finite-atom tail. Otherwise return null for the maximum; no finite index certifies all positive atoms. This finite-atom maximum is not an arbitrary infinite-space argument. The exact result, retaining the requested order, is ([[0, 1/2, 25, Yes, Yes], [1, 1/2, 26, Yes, Yes]], 26, Yes, uniform_finite_atom_tail_after_maximum)."},"skills":["A Finite Number of Positive Atoms Gives a Maximum Settling Index"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0051","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e62c52ef55862aec825914b2ee07be63c7e958b8a726d19749034e8e80e6904","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014051,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[1,2,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[1, 2, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0052","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"052e928fe6cf627602b30b9a174734ac9b2322b3036a9c7f8f5f21b7f558a1b7","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014052,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[2,3,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[2, 3, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0053","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7d367dbb4323b8d19c48e2db37b0ee865ffb31e4b01fd4acceea21df8edd88b","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014053,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[3,4,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[3, 4, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0054","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4a1d7a11ae8a4a1d822a0ffe7bda49a6eefbef3d7ff5076fcadec9f6418da9c","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014054,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[4,5,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[4, 5, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0055","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15dee1002b400fc05ae51bc2c14bb25083776d16a9eb163d5144e315a2995c1a","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014055,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[5,6,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[5, 6, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0056","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a1315b077ca967d03c996a15b5465144ea1201b3bc83e621330a1d850714623","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014056,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[6,7,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[6, 7, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0057","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ece63eeee612d9557bfdb2218cfb115b7385127a0f3cfd3b1793501c6801223e","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014057,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[7,8,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[7, 8, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0058","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"39efb71f9078d149d0f0f776d35b1b17c0860fbfe769b1dac994af2eede8f671","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014058,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[8,9,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[8, 9, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0059","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"321e3b9174cef454e104e2a0756fc02418f29c0cba16f7d7c9a46618b394d268","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014059,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[9,10,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[9, 10, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0060","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d0a5a88525a70bc84dd4c0a2486c9e4280431177d4937a64cdcd141bef2fcd3","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014060,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[10,11,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[10, 11, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0061","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82b5ae20540a5c7cf4924eb6567b3b2c190bf2944f59ac024d39efee63008ace","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014061,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[11,12,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[11, 12, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0062","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6197ed84e420288bfa643bc3fecda3cba88af3d6e377e521f363595027d5a7c6","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014062,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[12,13,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[12, 13, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0063","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c643354a336719cc48f8ca0c773a1c915ddf1e1db19717a20930948baca4fc8","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014063,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[13,14,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[13, 14, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0064","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90f98a8957b7d4c318aa99e088b44a89106e71c8eb660e7186b305c18c5f6748","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014064,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[14,15,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[14, 15, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0065","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6e310cac9c31e8c53cee185c89b9e8e5328c4b59a198d848bb34a64752b386d","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014065,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[15,16,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[15, 16, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0066","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c492e34151402a87be89479c982067b288e12c53035dbc2430882e5e93e38427","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014066,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[16,17,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[16, 17, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0067","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e92dfcb1f30b134a2e48eb57848f74c7a901137d36b98cde6cb2a44e3e0d0508","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014067,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[17,18,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[17, 18, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0068","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b857092f91496dd9bdf4b6ab5e9b38ff81d9ceaae1b87e1dd1d73c162deeb26b","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014068,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[18,19,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[18, 19, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0069","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8844db1ed439218c09bf1ba16be7cbe67afc456af1c4212843b01851e8238782","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014069,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[19,20,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[19, 20, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0070","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c72b9047d61704fd2f0d0b07cec3da116312a2d3a047d7b2bfd94e78e0805a5","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014070,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[20,21,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[20, 21, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0071","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78ca3aa40ee4aaa64cebb8cdc6219005966273d2f70d564711010ad3945504d6","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014071,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[21,22,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[21, 22, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0072","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46f720e173982fa2c2d3f65ca587ac5d966db1e4614fb01984a1d3adb3d95dbc","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014072,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[22,23,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[22, 23, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0073","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9201dbfd050c790c363ee77c0a9fbeb5f0e28f08117d985895007ebe7c28486","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014073,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[23,24,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[23, 24, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0074","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bda9f7069b01113015f2d5a284d1fa6e53879403b4bdaca1f5c445dd9b817560","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014074,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[24,25,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[24, 25, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0075","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"66321bb46f6b726c1e715cbc9a59e0445b1ff3c18565737356ba3aba284e9278","family_id":"qf-mod32-asconv-zero-mass-boundary","seed":202611014075,"semantic_template":"qf-mod32-asconv-zero-mass-boundary/{\"atom\":2,\"masses\":[[1,2],[1,2],[0,1]],\"settling_indices\":[25,26,null]}"},"fields":{"title":"A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[25, 26, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Selected atom=2 must actually have zero mass. Give (atom,mass,settling_index,ignored?,almost_sure?,zero_mass_failure_ignored_only_for_almost_sure_claim).","answer":{"kind":"expression","value":"(2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)","explanation":"Check the selected mass before ignoring its failure for the almost-sure test. Keep its settling declaration visible and still audit every positive-mass atom. This does not make a false pointwise statement true on the null atom. The exact result, retaining the requested order, is (2, 0, null, Yes, Yes, zero_mass_failure_ignored_only_for_almost_sure_claim)."},"skills":["A Null Atom Can Fail Pointwise without Changing the Almost-Sure Claim"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0076","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7446e1a08a39cd23b06373c1b596c535830764c9e59ebceeabde2beff2837515","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014076,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":1,\"settling_indices\":[1,2,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[1, 2, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=1. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(1, [[0, 1/2, 1, Yes, Yes], [1, 1/2, 2, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (1, [[0, 1/2, 1, Yes, Yes], [1, 1/2, 2, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0077","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f2d26bd7f9527ba2a195d5ea030cb0777eef206c302002a295bb81a13ff52e2","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014077,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":2,\"settling_indices\":[2,3,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[2, 3, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=2. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(2, [[0, 1/2, 2, Yes, Yes], [1, 1/2, 3, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (2, [[0, 1/2, 2, Yes, Yes], [1, 1/2, 3, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0078","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cb72f8da1b30e67de42f6961f8fd2bedcafdbb6df906f40147255d09da2a716","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014078,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":3,\"settling_indices\":[3,4,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[3, 4, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=3. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(3, [[0, 1/2, 3, Yes, Yes], [1, 1/2, 4, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (3, [[0, 1/2, 3, Yes, Yes], [1, 1/2, 4, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0079","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"483e343921d81dabfa3217bc034142cd9b8d4afa4482a9dce35b4e2313d6ba41","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014079,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":4,\"settling_indices\":[4,5,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[4, 5, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=4. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(4, [[0, 1/2, 4, Yes, Yes], [1, 1/2, 5, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (4, [[0, 1/2, 4, Yes, Yes], [1, 1/2, 5, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0080","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74dec95bf6a17534067468024f62ed574b045e068f3821f44f7c75642b54eed4","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014080,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":5,\"settling_indices\":[5,6,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[5, 6, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=5. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(5, [[0, 1/2, 5, Yes, Yes], [1, 1/2, 6, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (5, [[0, 1/2, 5, Yes, Yes], [1, 1/2, 6, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0081","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c092e21f054dd86fb0c0e1e73fcbc04a6d2b5bd5923898d18f678a1c260ec10","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014081,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":6,\"settling_indices\":[6,7,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[6, 7, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=6. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(6, [[0, 1/2, 6, Yes, Yes], [1, 1/2, 7, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (6, [[0, 1/2, 6, Yes, Yes], [1, 1/2, 7, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0082","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ce956d53da0eb44aa573cf0c644b964d272e1b8ca141d661b8aa034af6af6c7","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014082,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":7,\"settling_indices\":[7,8,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[7, 8, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=7. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(7, [[0, 1/2, 7, Yes, Yes], [1, 1/2, 8, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (7, [[0, 1/2, 7, Yes, Yes], [1, 1/2, 8, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0083","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"978605e730a94a77b5a262a85ababbec3e08675adf1259d46037001ea80c72fe","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014083,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":8,\"settling_indices\":[8,9,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[8, 9, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=8. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(8, [[0, 1/2, 8, Yes, Yes], [1, 1/2, 9, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (8, [[0, 1/2, 8, Yes, Yes], [1, 1/2, 9, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0084","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d478e76f1fc0bded615466a973d9d02af1ab1609983234c9fb692812b711d8c6","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014084,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":9,\"settling_indices\":[9,10,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[9, 10, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=9. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(9, [[0, 1/2, 9, Yes, Yes], [1, 1/2, 10, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (9, [[0, 1/2, 9, Yes, Yes], [1, 1/2, 10, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0085","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7cdfef2b8be080e06f28ef28e456f43e884f37861528e1499841807a4fb1d64","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014085,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":10,\"settling_indices\":[10,11,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[10, 11, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=10. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(10, [[0, 1/2, 10, Yes, Yes], [1, 1/2, 11, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (10, [[0, 1/2, 10, Yes, Yes], [1, 1/2, 11, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0086","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da401f6c4c681c26bdd790d2b84348f19b28cca27dcbc6547be5e4619c789e75","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014086,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":11,\"settling_indices\":[11,12,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[11, 12, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=11. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(11, [[0, 1/2, 11, Yes, Yes], [1, 1/2, 12, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (11, [[0, 1/2, 11, Yes, Yes], [1, 1/2, 12, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0087","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"26882ae5feca422f4a8ff01768935ba81673a43194f48a4e5d8a65736cf91338","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014087,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":12,\"settling_indices\":[12,13,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[12, 13, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=12. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(12, [[0, 1/2, 12, Yes, Yes], [1, 1/2, 13, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (12, [[0, 1/2, 12, Yes, Yes], [1, 1/2, 13, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0088","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd8bcebc7beb17eefd1afae42e26287239117f2e459fe318450dfbaa97882625","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014088,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":13,\"settling_indices\":[13,14,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[13, 14, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=13. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(13, [[0, 1/2, 13, Yes, Yes], [1, 1/2, 14, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (13, [[0, 1/2, 13, Yes, Yes], [1, 1/2, 14, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0089","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f96c7703ce4206626b05065bdb80c2abfd5ee0252feb8ed4061cd366198f3a65","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014089,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":14,\"settling_indices\":[14,15,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[14, 15, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=14. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(14, [[0, 1/2, 14, Yes, Yes], [1, 1/2, 15, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (14, [[0, 1/2, 14, Yes, Yes], [1, 1/2, 15, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0090","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5e919686d367cdf9bcff0e598718e9ebd9aaa7cf5b5c52295bed3d2613989da","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014090,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":15,\"settling_indices\":[15,16,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[15, 16, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=15. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(15, [[0, 1/2, 15, Yes, Yes], [1, 1/2, 16, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (15, [[0, 1/2, 15, Yes, Yes], [1, 1/2, 16, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0091","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f1e36c0f8e6f879e020f91ac381e2d1811a721b0b2c88c1ea04e94838b39193","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014091,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":16,\"settling_indices\":[16,17,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[16, 17, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=16. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(16, [[0, 1/2, 16, Yes, Yes], [1, 1/2, 17, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (16, [[0, 1/2, 16, Yes, Yes], [1, 1/2, 17, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0092","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4357f536b1392b521ec650fdfbb1d2b47c8baa4f54942c56433de0419831def7","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014092,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":17,\"settling_indices\":[17,18,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[17, 18, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=17. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(17, [[0, 1/2, 17, Yes, Yes], [1, 1/2, 18, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (17, [[0, 1/2, 17, Yes, Yes], [1, 1/2, 18, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0093","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3dc40a7279d725faf8f0ffa42d049ec8512b81fd7941114f967ad7f4e3eea06","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014093,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":18,\"settling_indices\":[18,19,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[18, 19, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=18. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(18, [[0, 1/2, 18, Yes, Yes], [1, 1/2, 19, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (18, [[0, 1/2, 18, Yes, Yes], [1, 1/2, 19, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0094","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03d3e9f39733cca012d3d8b9447174affcda28889682b4d135f9a81d4d96dcce","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014094,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":19,\"settling_indices\":[19,20,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[19, 20, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=19. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(19, [[0, 1/2, 19, Yes, Yes], [1, 1/2, 20, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (19, [[0, 1/2, 19, Yes, Yes], [1, 1/2, 20, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0095","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d367bdfaa52e2e7df111d170ae1a73e1fb0898682d1687c2f5a96d2927ac0d44","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014095,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":20,\"settling_indices\":[20,21,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[20, 21, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=20. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(20, [[0, 1/2, 20, Yes, Yes], [1, 1/2, 21, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (20, [[0, 1/2, 20, Yes, Yes], [1, 1/2, 21, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0096","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22b980054a71a5a0a568c865e7359c6f56004cc7418effc84d4a9dddd7cdeb3a","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014096,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":21,\"settling_indices\":[21,22,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[21, 22, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=21. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(21, [[0, 1/2, 21, Yes, Yes], [1, 1/2, 22, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (21, [[0, 1/2, 21, Yes, Yes], [1, 1/2, 22, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0097","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15ce026d346e2aed200c1715060c9d8db9d81a67ea33fbd5e226b7add2661cf3","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014097,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":22,\"settling_indices\":[22,23,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[22, 23, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=22. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(22, [[0, 1/2, 22, Yes, Yes], [1, 1/2, 23, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (22, [[0, 1/2, 22, Yes, Yes], [1, 1/2, 23, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0098","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef30eb7584175adbe7f732d4b36c1dc4864eea067b7c1cb3394032b1ee72909e","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014098,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":23,\"settling_indices\":[23,24,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[23, 24, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=23. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(23, [[0, 1/2, 23, Yes, Yes], [1, 1/2, 24, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (23, [[0, 1/2, 23, Yes, Yes], [1, 1/2, 24, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0099","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76e85410828b3740d1190a169b041ee4e5b52a7642850af7135b7c2e81cd6861","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014099,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":24,\"settling_indices\":[24,25,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[24, 25, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=24. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(24, [[0, 1/2, 24, Yes, Yes], [1, 1/2, 25, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (24, [[0, 1/2, 24, Yes, Yes], [1, 1/2, 25, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261101-r04-almost-sure-convergence-0100","batch_id":"batch-20261014-advance-20261101-r04-almost-sure-convergence-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddc544d3bd488b8e8cde416a28e6ff7a4b02865df55cbe08cd9b2153cbeceb51","family_id":"qf-mod32-asconv-finite-prefix-boundary","seed":202611014100,"semantic_template":"qf-mod32-asconv-finite-prefix-boundary/{\"masses\":[[1,2],[1,2],[0,1]],\"prefix_length\":25,\"settling_indices\":[25,26,null]}"},"fields":{"title":"A Finite Prefix Alone Does Not Establish Every Atomwise Tail","stem":"Use one complete finite atomic probability space with aligned exact nonnegative masses=[1/2, 1/2, 0/1], summing to 1, and supplied full-tail settling_indices=[25, 26, null]. An integer N declares equality to the common limit from that index onward on this atom; null denotes a model-declared nonconvergent atom, not merely a missing file or unchecked proof receipt. Every positive-mass atom must converge; a zero-mass failure is ignored only for the almost-sure claim. Keep exact reduced rational text, omitting /1 for integers. A finite observed prefix alone is not this universal-tail certificate. Observed prefix_length=25. Give (prefix_length,atom_rows,sufficient?,finite_prefix_not_almost_sure_convergence_certificate).","answer":{"kind":"expression","value":"(25, [[0, 1/2, 25, Yes, Yes], [1, 1/2, 26, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)","explanation":"The No sufficiency flag refers to finite-prefix evidence only, not to refutation of a separately declared full-tail certificate in the rows. General convergence-mode warnings must not erase the explicit common finite-atom assumptions or confuse a missing observation with a nonconvergent-atom declaration. The exact result, retaining the requested order, is (25, [[0, 1/2, 25, Yes, Yes], [1, 1/2, 26, Yes, Yes], [2, 0, null, No, Yes]], No, finite_prefix_not_almost_sure_convergence_certificate)."},"skills":["A Finite Prefix Alone Does Not Establish Every Atomwise Tail"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}